-
Notifications
You must be signed in to change notification settings - Fork 23
/
BlakersMassey.ard
234 lines (199 loc) · 14.2 KB
/
BlakersMassey.ard
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
{- | The generalized Blakers-Massey theorem.
- The proof combines ideas from the following papers:
- * Kuen-Bang Hou (Favonia), Eric Finster, Dan Licata, Peter LeFanu Lumsdaine, A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory (https://arxiv.org/abs/1605.03227, https://github.com/HoTT/HoTT-Agda/blob/master/theorems/homotopy/BlakersMassey.agda)
- * Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal, A Generalized Blakers-Massey Theorem (https://arxiv.org/abs/1703.09050)
- -}
\import Equiv
\import Equiv.Sigma
\import Equiv.Univalence
\import HLevel
\import Homotopy.Fibration
\import Homotopy.Join
\import Homotopy.Localization.Connected
\import Homotopy.Localization.Universe
\import Homotopy.Pushout
\import Logic
\import Paths
\import Paths.Meta
\lemma genBlakersMassey {d : Data} (x0 : X) (y0 : Y) : isConnectedMap (pbMap {d} {x0} {y0})
=> \lam p => surjective (POData.pullback_pushout-surjective {\new POData Y X (\lam y x => Q x y)} y0 x0 (inv (pmap swap p))) p
\where {
\open Localization
-- | It is easier to prove the theorem if the type ``\Sigma (y : Y) (Q x0 y)`` is inhabited.
\lemma surjective {d : Data} {x0 : X} {y0 : Y} (q : TruncP (\Sigma (y : Y) (Q x0 y))) : isConnectedMap (pbMap {d} {x0} {y0}) \elim q
| inP (y,q0) => \lam p => contr=>isConnected (Fib pbMap p) (DataExt.code-contr {\new DataExt { | Data => d | x0 => x0 | y0 => y | q0 => q0 }} p)
\class EquivData \noclassifying (A B : \hType) \extends ReflUniverse {
| M : A -> Connected
| N : B -> Connected
| f : \Sigma (a : A) (M a) -> \Sigma (b : B) (N b)
| g : \Sigma (b : B) (N b) -> \Sigma (a : A) (M a)
| p : \Pi (a : A) (m : M a) -> (g (f (a,m))).1 = a
| q : \Pi (b : B) (n : N b) -> (f (g (b,n))).1 = b
\func eval (a : A) (m : M a) : lift (\lam a => sec {Connected.equiv {M a} (LType B)} (\lam m => lEta (f (a,m)).1)) (lEta a) = lEta (f (a,m)).1 =>
lift-prop (\lam a => sec {Connected.equiv {M a} (LType B)} (\lam m => lEta (f (a,m)).1)) a *> path (\lam i => (f_sec {Connected.equiv {M a} (LType B)} (\lam m => lEta (f (a,m)).1) @ i) m)
\lemma equiv-lemma : Equiv {LType A} {LType B} (lift (\lam a => sec {Connected.equiv {M a} (LType B)} (\lam m => lEta (f (a,m)).1)))
=> \let | E1 a => Connected.equiv {M a} (LType B)
| E2 b => Connected.equiv {N b} (LType A)
| f1 a m => lEta (f (a,m)).1
| g1 b n => lEta (g (b,n)).1
| F a => sec {E1 a} (f1 a)
| G b => sec {E2 b} (g1 b)
\in localization-equiv F G
(\lam a => sec {Connected.equiv {M a} (pathLocal (lift G (F a)) (lEta a))}
(\lam m => lift G (F a) ==< path (\lam i => lift G ((f_sec {E1 a} (f1 a) @ i) m)) >==
lift G (f1 a m) ==< lift-prop G (f (a,m)).1 >==
G (f (a,m)).1 ==< path (\lam i => (f_sec {E2 (f (a,m)).1} (g1 (f (a,m)).1) @ i) (f (a,m)).2) >==
inL (g (f (a,m))).1 ==< pmap inL (p a m) >==
inL a `qed))
(\lam b => sec {Connected.equiv {N b} (pathLocal (lift F (G b)) (lEta b))}
(\lam n => lift F (G b) ==< path (\lam i => lift F ((f_sec {E2 b} (g1 b) @ i) n)) >==
lift F (g1 b n) ==< lift-prop F (g (b,n)).1 >==
F (g (b,n)).1 ==< path (\lam i => (f_sec {E1 (g (b,n)).1} (f1 (g (b,n)).1) @ i) (g (b,n)).2) >==
inL (f (g (b,n))).1 ==< pmap inL (q b n) >==
inL b `qed))
}
\class POData {
| X : \hType
| Y : \hType
| Q : X -> Y -> \hType
\func PO => PushoutData {\Sigma (x : X) (y : Y) (Q x y)} __.1 __.2
\func swap (x : PO) : PushoutData {\Sigma (y : Y) (x : X) (Q x y)} __.1 __.2
| pinl b => pinr b
| pinr c => pinl c
| pglue (x,y,q) i => pglue (y,x,q) (inv (path (\lam j => j)) @ i)
\func pbMap {x : X} {y : Y} (q : Q x y) : pinl x = {PO} pinr y => PushoutData.ppglue ((x,y,q) : \Sigma (x : X) (y : Y) (Q x y))
\lemma pullback_pushout-surjective (x : X) (y : Y) (p : pinl x = {PO} pinr y) : TruncP (\Sigma (x' : X) (Q x' y))
=> \have | EP => \new EmbeddingPushout {
| A => \Sigma (y : Y) (TruncP (\Sigma (x : X) (Q x y)))
| B => PushoutData {\Sigma (x : X) (y : Y) (Q x y)} {X} {\Sigma (y : Y) (TruncP (\Sigma (x : X) (Q x y)))} __.1 (\lam p => (p.2, inP (p.1,p.3)))
| C => Y
| f => pinr
| g => Embedding.projection (\lam y => TruncP (\Sigma (x : X) (Q x y)))
}
| t => ret {EP.pullback-path-equiv (pinl x) y} (pmap
(\case __ \with {
| pinl x => pinl (pinl x)
| pinr y => pinr y
| pglue (x,y,q) i =>
(pmap pinl (PushoutData.ppglue {\Sigma (x : X) (y : Y) (Q x y)} (x,y,q)) *>
PushoutData.ppglue {EP.A} (y, inP (x,q))) @ i
}) p)
\in rewrite t.3 t.1.2
}
\open POData
\class Data \extends ReflUniverse, POData
| ch {x x' : X} {y y' : Y} (q0 : Q x y) (q1 : Q x y') (q2 : Q x' y)
: isConnectedType (Join (\Sigma (p : y = y') (transport (Q x) p q0 = q1)) (\Sigma (p : x = x') (transport (`Q y) p q0 = q2)))
\class DataExt \extends Data {
| x0 : X
| y0 : Y
| q0 : Q x0 y0
\func code-left {x : X} {w : PO} (p : pinl x = {PO} w) (p' : pinl x0 = w) : \hType
=> LType (\Sigma (q1 : Q x y0) (pbMap q0 *> inv (pbMap q1) *> p = p'))
\func code-glue-gen {x : X} {w : PO} (code : pinl x0 = w -> \hType) (p : pinl x = w)
(t : \Pi (p' : pinl x0 = w) -> Equiv {code-left p p'} {code p'})
: transport (pinl x0 = __ -> \hType) p (code-left idp) = code \elim p
| idp => path (\lam i p' => Equiv-to-= (t p') @ i)
\func code-glue {x : X} {y : Y} (q : Q x y)
: transport (pinl x0 = __ -> \hType) (pbMap q) (code-left idp) = (\lam p => LType (Fib pbMap p))
=> code-glue-gen (\lam p => LType (Fib pbMap p)) (pbMap q) (code.equiv q)
\func code {w : PO} (p : pinl x0 = w) : \hType \elim w
| pinl x => code-left idp p
| pinr y => LType (Fib pbMap p)
| pglue (x,y,q) i => (pathOver (code-glue q) @ i) p
\where {
\func equivData {x : X} {y : Y} (q : Q x y) (p : pinl x0 = {PO} pinr y) : EquivData \cowith
| isLocal => isLocal
| localization => localization
| A => \Sigma (q1 : Q x y0) (pbMap q0 *> inv (pbMap q1) *> pbMap q = p)
| B => Fib pbMap p
| M a => \new Connected {
| X => Join (\Sigma (p : y0 = y) (transport (Q x) p a.1 = q)) (\Sigma (p : x = x0) (transport (`Q y0) p a.1 = q0))
| connected => ch a.1 q q0
}
| N b => \new Connected {
| X => Join (\Sigma (p : y = y0) (transport (Q x0) p b.1 = q0)) (\Sigma (p : x0 = x) (transport (`Q y) p b.1 = q))
| connected => ch b.1 q0 q
}
| f ad => LR q p ad.1 ad.2
| g bd => RL q0 q p bd.1 bd.2
| p => LRL q p
| q => RLR q p
\lemma equiv {x : X} {y : Y} (q : Q x y) (p : pinl x0 = {PO} pinr y) => EquivData.equiv-lemma {equivData q p}
\func pathLem1 {A : \hType} {a a' a'' : A} (p : a = a') (q : a'' = a') : p *> inv q *> q = p
| idp, idp => idp
\func pathLem2-gen {A : \hType} {a a' a'' : A} {p1 p2 : a = a'} {q : a = a''} (t : p1 *> inv p2 = idp) : p1 *> inv p2 *> q = q \elim q
| idp => t
\func pathLem2 {A : \hType} {a a' a'' : A} (p : a = a') (q : a = a'') : p *> inv p *> q = q
=> pathLem2-gen (*>_inv p)
\func pathLem3 {A : \hType} {a a' : A} (p : a = a') : pathLem1 p p = pathLem2 p p
| idp => idp
\func LR {x : X} {y : Y} (q : Q x y) (p : pinl x0 = {PO} pinr y)
(a : \Sigma (q1 : Q x y0) (pbMap q0 *> inv (pbMap q1) *> pbMap q = p))
(m : Join (\Sigma (p : y0 = y) (transport (Q x) p a.1 = q)) (\Sigma (p : x = x0) (transport (`Q y0) p a.1 = q0)))
: \Sigma (b : Fib pbMap p) (Join (\Sigma (p : y = y0) (transport (Q x0) p b.1 = q0)) (\Sigma (p : x0 = x) (transport (`Q y) p b.1 = q)))
\elim a, m
| (a1,a2), jinl (idp,idp) => ((q0, inv (pathLem1 (pbMap q0) (pbMap a1)) *> a2), pinl (idp,idp))
| (a1,a2), jinr (idp,idp) => ((q, inv (pathLem2 (pbMap a1) (pbMap q)) *> a2), pinr (idp,idp))
| (a1,a2), pglue ((idp,idp),(idp,idp)) i => ((q0, inv (pathLem3 (pbMap q0) @ i) *> a2), pglue ((idp,idp),(idp,idp)) i)
\func RL {x : X} {y : Y} (q0 : Q x0 y0) (q : Q x y) (p : pinl x0 = {PO} pinr y)
(b : \Sigma (q2 : Q x0 y) (pbMap q2 = p))
(n : Join (\Sigma (p : y = y0) (transport (Q x0) p b.1 = q0)) (\Sigma (p : x0 = x) (transport (`Q y) p b.1 = q)))
: \Sigma (a : \Sigma (q1 : Q x y0) (pbMap q0 *> inv (pbMap q1) *> pbMap q = p))
(Join (\Sigma (p : y0 = y) (transport (Q x) p a.1 = q)) (\Sigma (p : x = x0) (transport (`Q y0) p a.1 = q0)))
\elim n
| jinl (idp,idp) => ((q, pathLem1 (pbMap b.1) (pbMap q) *> b.2), pinl (idp,idp))
| jinr (idp,idp) => ((q0, pathLem2 (pbMap q0) (pbMap b.1) *> b.2), pinr (idp,idp))
| pglue ((idp,idp),(idp,idp)) i => ((b.1, (pathLem3 (pbMap b.1) @ i) *> b.2), pglue ((idp,idp),(idp,idp)) i)
\func LRL {x : X} {y : Y} (q : Q x y) (p : pinl x0 = {PO} pinr y)
(a : \Sigma (q1 : Q x y0) (pbMap q0 *> inv (pbMap q1) *> pbMap q = p))
(m : Join (\Sigma (p : y0 = y) (transport (Q x) p a.1 = q)) (\Sigma (p : x = x0) (transport (`Q y0) p a.1 = q0)))
: (RL q0 q p (LR q p a m).1 (LR q p a m).2).1 = a
\elim a, m
| (a1,idp), jinl (idp,idp) => path (\lam i => (q, *>_inv (pathLem1 (pbMap q0) (pbMap q)) @ i))
| (a1,idp), jinr (idp,idp) => path (\lam i => (q0, *>_inv (pathLem2 (pbMap q0) (pbMap q)) @ i))
| (a1,idp), pglue ((idp,idp),(idp,idp)) j => path (\lam i => (q0, *>_inv (pathLem3 (pbMap q0) @ j) @ i))
\func RLR {x : X} {y : Y} (q : Q x y) (p : pinl x0 = {PO} pinr y)
(b : \Sigma (q2 : Q x0 y) (pbMap q2 = p))
(n : Join (\Sigma (p : y = y0) (transport (Q x0) p b.1 = q0)) (\Sigma (p : x0 = x) (transport (`Q y) p b.1 = q)))
: (LR q p (RL q0 q p b n).1 (RL q0 q p b n).2).1 = b
\elim b, n
| (b1,idp), jinl (idp,idp) => path (\lam i => (q0, inv_*> (pathLem1 (pbMap q0) (pbMap q)) @ i))
| (b1,idp), jinr (idp,idp) => path (\lam i => (q, inv_*> (pathLem2 (pbMap q0) (pbMap q)) @ i))
| (b1,idp), pglue ((idp,idp),(idp,idp)) j => path (\lam i => (q0, inv_*> (pathLem3 (pbMap q0) @ j) @ i))
}
\func coerce-path-gen {w : PO} (p : pinl x0 = w) (t : transport (pinl x0 = __ -> \hType) p (code-left idp) = code)
: code-left idp idp = code p \elim p
| idp => path ((t @ __) idp)
\func coerce-path {w : PO} (p : pinl x0 = w)
: code-left idp idp = code p
=> coerce-path-gen p (pmapd (\lam _ => code) p)
\func Left {w : PO} (p : pinl x0 = w) => \Sigma (q : Q x0 y0) (pbMap q0 *> inv (pbMap q) *> p = p)
\func code-left-diag {w : PO} (p : pinl x0 = w) (v : code-left idp idp) : code-left p p
=> lmap {_} {Left idp} {Left p} (\lam t => (t.1, code.pathLem2-gen t.2)) v
\func coerce-path-glue-gen {w : PO} (p : pinl x0 = w) (t : \Pi (p' : pinl x0 = w) -> Equiv {code-left p p'} {code p'}) (v : code-left idp idp)
: transport (\lam T => T) (coerce-path-gen p (code-glue-gen code p t)) v = t p (code-left-diag p v) \elim p
| idp => pmap (t idp) (inv (lmap.id-prop v))
\func code-center {w : PO} (p : pinl x0 = w) : code p
=> transport (\lam T => T) (coerce-path p) (lEta point)
\where \func point : Left idp => (q0, *>_inv (pbMap q0))
\func code-path (p : pinl x0 = {PO} pinr y0) (c : Fib pbMap p) : code-center p = lEta c \elim c
| (q,idp) =>
code-center (pbMap q)
==< pmap (\lam s => transport (\lam T => T) (coerce-path-gen (pbMap q) s) (lEta code-center.point)) (pmapd_pathOver (pinl x0 = __ -> \hType) (\lam _ => code) (pbMap q) (code-glue q) idp) >==
transport (\lam T => T) (coerce-path-gen (pbMap q) (code-glue q)) (lEta code-center.point)
==< coerce-path-glue-gen (pbMap q) (code.equiv q) (lEta code-center.point) >==
code.equiv q (pbMap q) (code-left-diag (pbMap q) (lEta code-center.point))
==< pmap (code.equiv q (pbMap q)) (lift-prop (\lam (t : Left idp) => inL ((t.1, code.pathLem2-gen t.2) : Left (pbMap q))) code-center.point) >==
code.equiv q (pbMap q) (lEta ((q0, code.pathLem2 (pbMap q0) (pbMap q)) : Left (pbMap q)))
==< EquivData.eval {code.equivData q (pbMap q)} (q0, code.pathLem2 (pbMap q0) (pbMap q)) (pinr (idp,idp)) >==
lEta ((q, inv (code.pathLem2 (pbMap q0) (pbMap q)) *> code.pathLem2 (pbMap q0) (pbMap q)) : Fib pbMap p)
==< pmap (\lam x => lEta ((q,x) : Fib pbMap p)) (inv_*> (code.pathLem2 (pbMap q0) (pbMap q))) >==
lEta ((q, idp) : Fib pbMap p)
`qed
\lemma code-contr {w : PO} (p : pinl x0 = w) : Contr (code p) \cowith
| center => code-center p
| contraction => transport (\lam t => \Pi (c : code t.2) -> code-center t.2 = c) (isContr=>isProp (lsigma {PO} (pinl x0)) (pinr y0, pbMap q0) (w, p))
(\lam c => remove_inL (\lam _ => code-center (pbMap q0)) (\lam c => c) (code-path (pbMap q0)) c)
}
}